$ 0.\overline{87} \div -3.\overline{17} = {?} $
Solution: First convert the repeating decimals to fractions. $\begin{align*} 100x &= 87.8787...\\ x &= 0.8787...\end{align*} $ $\begin{align*} 99x &= 87 \\ x &= \dfrac{87}{99}\end{align*} $ $\begin{align*} 100y &= -317.1718...\\ y &= -3.1718...\end{align*} $ $\begin{align*} 99y &= -314 \\ y &= -\dfrac{314}{99}\end{align*} $ So, the problem becomes: $ \dfrac{87}{99} \div -\dfrac{314}{99} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ \dfrac{87}{99} \times -\dfrac{99}{314} = {?} $ $ \phantom{\dfrac{87}{99} \times -\dfrac{314}{99}} = \dfrac{87 \times 99}{99 \times -314} $ $ \phantom{\dfrac{87}{99} \times -\dfrac{314}{99}} = \dfrac{87 \times \cancel{99}} {\cancel{99} \times -314} $ $ \phantom{\dfrac{87}{99} \times -\dfrac{314}{99}} = -\dfrac{87}{314} $